Projective special linear group is simple

This article gives the statement, and possibly proof, of a particular group or type of group (namely, Projective special linear group (?)) satisfying a particular group property (namely, Simple group (?)).

Statement

Let $k$ be a field and $n$ be a natural number greater than $1$. Then, the projective special linear group $PSL_{n}(k)$ is a simple group provided one of these conditions holds:

• $n\geq 3$.
• $k$ has at least four elements.

Facts used

1. Special linear group is perfect: Under the same conditions ($n\geq 3$ or $k$ has at least four elements), the special linear group $SL_{n}(k)$ is a perfect group: it equals its own derived subgroup.
2. Perfectness is quotient-closed: The quotient of a perfect group by a normal subgroup is perfect.
3. Abelian normal subgroup of core-free maximal subgroup is contranormal implies derived subgroup of whole group is monolith

Related facts

Related facts about special linear group and projective special linear group

• Special linear group is perfect
• Special linear group is quasisimple
• Projective special linear group equals alternating group in only finitely many cases

Related facts about simplicity of linear groups

• Projective symplectic group is simple
• Projective special orthogonal group for bilinear form of positive Witt index is simple
• projective special orthogonal group over reals is simple

Proof

The proof proceeds in the following steps:

1. $PSL_{n}(k)$ satisfies the hypotheses for fact (3): Consider the natural action of $PSL_{n}(k)$ on the projective space $\mathbb {P} ^{n-1}(k)$. This is a primitive group action, and the stabilizer of any point is thus a core-free maximal subgroup. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
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2. The commutator subgroup of $PSL_{n}(k)$ is contained in every nontrivial normal subgroup of $PSL_{n}(k)$: This follows from the previous step and fact (3).
3. $PSL_{n}(k)$ equals its own commutator subgroup when $n\geq 3$ or $k$ has at least four elements: This follows from facts (1) and (2).
4. $PSL_{n}(k)$ is simple when $n\geq 3$ or $k$ has at least four elements: : This follows from the last two steps.